THE FUNDAMENTAL STRUCTURE OF THE WORLD: PHYSICAL MAGNITUDES, SPACE AND TIME, AND THE LAWS OF NATURE MARCO KORSTIAAN DEES

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1 THE FUNDAMENTAL STRUCTURE OF THE WORLD: PHYSICAL MAGNITUDES, SPACE AND TIME, AND THE LAWS OF NATURE By MARCO KORSTIAAN DEES A dissertation submitted to the Graduate School-New Brunswick Rutgers, The State University of New Jersey In partial fulfillment of the requirements For the degree of Doctor of Philosophy Graduate Program in Philosophy Written under the direction of Professor Dean Zimmerman And approved by New Brunswick, New Jersey May 2015

2 ABSTRACT OF THE DISSERTATION The Fundamental Structure of the World: Physical Magnitudes, Space and Time, and the Laws of Nature By MARCO KORSTIAAN DEES Dissertation Director: Dean Zimmerman What is the world fundamentally like? In my dissertation I explore and defend the idea that we should look for accounts of reality that avoid redundant structure. This idea plays a central role in science, and I believe its has the potential to be extremely powerful and fruitful in metaphysics as well. I identify three forms of redundancy in metaphysics: empirical, metaphysical and axiomatic redundancy. Avoiding these forms of redundancy imposes powerful constraints on acceptable accounts in metaphysics; we should look for views that (i) do not posit unnecessary structure, (ii) characterize the world without redundancy, and (iii) avoid unexplained patterns at the fundamental level. I argue against widely accepted accounts of physical magnitudes and space and time on the basis that they suffer from these forms of explanatory redundancy, and in their place I develop novel accounts that are not explanatorily defective in this way. Chapter one argues that the structure of quantitative properties is reducible to facts about the dynamical roles different magnitudes play in the laws of nature, so that 2kg mass is greater than 1kg mass in virtue of the fact that these magnitudes give rise to different consequences for how things accelerate. Chapter two argues that the spatial and temporal arrangement of the world reduces to facts about its causal structure, so that you are closer to your pint of beer than to the moon in virtue of the ii.

3 fact that you causally interact more strongly with the beer than the moon. In the final chapter I argue that although physics describes the world in the language of mathematics, there are compelling reasons to think that this description is not fundamental, for it is extrinsic and involves conventional choices of scale. If this is right then corresponding to every mathematical description of the world there is an intrinsic description that characterizes the physical structure of reality directly. I conclude that the fundamental physical laws are not the laws of physics. iii.

4 ACKNOWLEDGEMENTS I could not have written this dissertation without the support, feedback, and criticism of many others. I received helpful comments from Bob Beddor, David Black, David Chalmers, Eddy Chen, Shamik Dasgupta, Tom Donaldson, Simon Goldstein, Ned Hall, Michael Hicks, Lucy Jordan, Martin Lin, Barry Loewer, Michaela McSweeney, Daniel Rubio, Jonathan Schaffer, Jason Stanley, Nick Tourville, Tobias Wilsch, Dean Zimmerman, and Jennifer Wang. I owe a particular intellectual debt to Tobias Wilsch. My time at Rutgers would not have been the same without the countless hours discussing and arguing about all kinds of weird and wonderful puzzles in metaphysics. Metaphysics just wouldn t have been as fun without him. I have been incredible fortunate to have such thoughtful, generous and, frankly, damn smart people on my dissertation committee: Ned Hall, Martin Lin, Barry Loewer, Jonathan Schaffer, and Dean Zimmerman. They were subjected to a lot of underdeveloped work and my dissertation is vastly better than it would have been without their help. I look up to each of them as models for how philosophy in its best form can be done. I am especially grateful to my advisors, Dean and Jonathan, for all their encouragement and support. I ve been lucky to work with people who are not just brilliant as philosophers but admirable as individuals besides. Thanks to my parents, Chrisjan and Anja Dees, for their truly unconditional love. Above all, my deepest gratitude is to Lucy Jordan. Lucy has shown me more kindness, generosity, and unshakable love than I deserve and has suffered more for the sake of me and the dissertation than I can ever make up for. I have learned from her something immeasurably more valuable than anything written about here. I learned how deep selfless love can go. I dedicate this dissertation to her. iv.

5 TABLE OF CONTENTS Title Page i. Dissertation Abstract Acknowledgements ii. iv. Table of Contents v. 1. Introduction 1 2. Physical Magnitudes 4 3. The Causal Theory of Spacetime Tim Maudlin on the Triangle Inequality The Fundamental Physical Laws are Not the Laws of Physics 70 v.

6 1 1. Introduction This dissertation is an attempt to tackle the question: What is the world fundamentally like? It is natural to think that in order to learn about the fundamental nature of the world we should look to our most fundamental science, physics. But it is far from clear what, exactly, we learn about the world from physics. In this dissertation I develop and defend the idea that we should look for accounts of reality that avoid explanatorily redundant structure. This idea plays a central role in science, and I believe its has the potential to be extremely powerful and fruitful in metaphysics as well. Very roughly, the idea is that we should look for accounts of reality that minimize what is left unexplained. I identify three kinds of redundancy in metaphysics that I call empirical, metaphysical, and axiomatic redundancy. Avoiding these forms of redundancy imposes powerful constraints on acceptable accounts in metaphysics; we should look for views that (i) do not posit unnecessary structure, (ii) characterize the world without redundancy, and (iii) avoid unexplained patterns at the fundamental level. These constraints are, I hope, extremely natural and independently attractive. Nonetheless, no mainstream package of views in metaphysics satisfies all three principles. I argue against widely accepted accounts of physical magnitudes and space and time on the basis that they suffer from these forms of explanatory redundancy, and in their place I develop novel accounts that are not explanatorily defective in this way. A theory contains empirically redundant structure if it posits things we dont need in order to make sense of the world. For example, just as we dont need to posit facts about witches to make sense of the world, I argue that we dont need to posit primitive facts about which things are more massive than others. A theory contains metaphysically redundant structure if it fails to satisfy a plausible

7 2 minimality constraint on the fundamental facts: we should attribute just enough structure to the world to characterize it completely, but no more. For example, once God had decreed that there be elephants, giraffes, and kangaroos, He didn t then need to decree that mammals exist this decree would be redundant. Many mainstream views in metaphysics fail to satisfy this natural requirement; for example, I argue against primitivism about spacetime on the grounds that there is no way for the primitivist to satisfy this constraint. A theory contains axiomatically redundant structure if makes unnecessary stipulations about how the basic building blocks of the world behave. We should prefer theories on which striking patterns in reality can be explained rather than simply taken as primitive. For example, I argue for a broadly structuralist account of physical quantities on the basis that this allows us to explain why nothing can have both 1kg mass and 2kg mass. On the standard approaches to physical quantities, constraints like this must simply be stipulated to hold. But brute patterns like these are mysterious, because it is natural to think that the basic building blocks of the world do not impose constraints on how they may be rearranged. The properties of physics are quantitative properties like mass and charge that come in determinate magnitudes like 1kg mass or 2kg mass. These magnitudes stand in a structure that allows us to make comparisons among them, such as the fact that 2 kg mass is greater than 1 kg mass. According to primitivism about quantity these comparisons are simply brute features of the world. The first chapter of the dissertation argues instead that quantitative relations between magnitudes reduce to facts about the dynamical effects of instantiating these magnitudes. So, for example, 2kg mass is greater than 1kg mass in virtue of what the laws say about the consequences of instantiating 2kg mass and 1kg mass. Both reductionists and non-reductionists about laws of nature typically take the structure of spacetime itself as primitive. The second chapter of my dissertation develops a view on which the spatial and temporal arrangement of the world is not primitive but instead reduces to facts about causal dependence. On this view you are closer to your coffee mug than to the moon in virtue of the fact that you can causally interact much more easily with your coffee mug than with the moon. A primary motivation for rejecting primitive facts about physical magnitudes and about space and time is that we can make sense of the world without them: there are simple,

8 3 elegant and explanatory accounts of the world that do not posit primitive facts about physical magnitudes or the structure of spacetime. The extra structure of the primitivist views is redundant structure. One worry about this line of argument is that it threatens to opens the door to skepticism, and so the reasoning to which it appeals cannot be sound. After all, one might think that we can fully account for patterns in our phenomenal states without positing any external world objects. But the argument from redundant structure differs from skeptical reasoning in three important ways. First, it is a claim about rational theory choice, about what we ought to believe, rather than about what we know or which of our beliefs are justified. Second, in the case of the external world there is no simple, elegant and explanatory competing theory capable of accounting for patterns in our phenomenal states. Third, it simply incredible that it is never rational to abandon one theory in favour of another on the basis that it attributes surplus structure to the world. This means there is a deep and general question about how to draw a distinction between skeptical reasoning and a sound avoidance of explanatorily redundant structure. I do not know the answer to this question; but every plausible candidate answer favours the application to physical magnitudes and the structure of spacetime. In the fourth chapter of my dissertation I argue that although physics describes the world in the language of mathematics, there are compelling reasons to think that this description is not fundamental, for it is extrinsic and involves conventional choices of scale. If this is right then corresponding to every mathematical description of the world there is an intrinsic description that characterizes the physical structure of reality directly. I conclude that the fundamental physical laws are not the laws of physics.

9 4 2. Physical Magnitudes Many familiar properties come in degrees. Hippos are heavier than hedgehogs, toads are taller than tadpoles, and flamingos fly faster than fleas. The properties of physics are also like this; electrons and protons have different magnitudes of charge, up quarks and down quarks have different spin, and x-rays and radio waves have different frequencies. Properties like these are quantities, and they have two characteristic features. A quantity like mass is associated with a family of determinate mass properties, like 1kg mass and 3.7kg mass. I ll say these are magnitudes of mass. Secondly, the magnitudes of a quantity stand in a structure that allows us to make comparisons among them, as when we say that the average African elephant is three times as massive as the average Rhinoceros. What is the metaphysical basis for comparisons like these? Answering this question is crucial to understanding the role quantities play in science. It is because quantities come in degrees that they are aptly represented by numbers, so that we can say that the elephant has a mass of 5,500kg. And this numerical representation is essential to laws of nature that state quantitative relationships between properties, like f = ma. Things behave the way they do because of the properties they have. So too for quantities; the more mass something has, the greater the gravitational attraction it experiences and the more it resists acceleration. What is the status of this claim? According to the mainstream view of physical quantities, quantity primitivism, this is just something that happens to be true given the form the laws take. 1 The fact that an elephant is more massive than an egg is not constituted or explained by the fact that I can lift only one of the two. In this paper I argue that this is a mistake. Instead, I will argue that facts about which things are more massive than others reduce to facts about which things give rise to greater gravitational 1 A version of quantity primitivism has been defended by almost everyone who has written about quantities, including Armstrong (1978), Bigelow and Pargetter (1988), Field (1980), and Mundy (1987).

10 5 attraction and resist acceleration more. On this view, elephants are more massive than eggs just in virtue of the fact that, say, elephants will crush things that eggs will not. I ll call this view nomic reductionism about quantity. I present two principal arguments for favouring nomic reductionism over quantity primitivism. The first is that quantity primitivism is committed to explanatorily redundant structure, and so we should prefer nomic reductionism on parsimony grounds. The second argument concerns the fact that magnitudes are not freely recombinable. Magnitudes in the same family are incompatible it is impossible for something to have both 1kg mass and 2kg mass. Necessary connections between properties call out for explanation, and whereas the nomic reductionism can provide an elegant explanation for this failure of free recombination, the quantity primitivist cannot. Here s the plan for the paper. Section 1 introduces nomic reductionism and quantity primitivism, and sections 2 and 3 present the argument from redundant structure and the argument from incompatibilities. The final section anticipates some objections. 1 Quantity Primitivism & Nomic Reductionism I will focus on the case of mass, although the issue I will raise arises for all quantities. The structure of mass allows us to make various kinds of mass comparisons. 2 For example, consider the following claims: (1) An elephant is more massive than an egg. (2) 1000kg is greater than 0.01kg Claims like (1) compare objects; they are first-order mass comparisons. Claims like (2) instead compare properties, so are second-order mass comparisons. One question that arises about quantities is whether first-order or second-order comparisons are prior. 3 2 I will focus on ordering relations among massive objects. But many physical quantities, including mass and charge, have more structure than this; they also have distance structure, so that it makes sense to say that 2kg mass is much closer in mass to 1kg mass than 1,000kg mass. I ll focus on ordering relations for simplicity. 3 Field (1980) offers a first-order account of mass. See Mundy (1989) for arguments in favour of secondorder over first-order accounts of quantity. The first-order/second-order distinction is related to the distinction between absolutist and comparativist accounts of quantity. A comparativist about quantity holds that

11 6 The issue I wish to raise in this paper is independent of the dispute between firstorder and second-order accounts of quantity. Instead, I want to ask whether some mass comparisons are fundamental, or whether they can all be explained in other terms. Quantity primitivism is the claim that there are fundamental comparisons among physical magnitudes, whether these are first-order or second-order comparisons. For example, Mundy (1987) invokes the two second-order relations, and. Intuitively, (p 1, p 2 ) encodes the fact that p 1 is less than or equal to p 2, and (p 1, p 2, p 3 ) encodes the fact that p 3 is the sum of p 1 and p 2. Field (1980) posits two first-order relations, mass-betweenness and mass-congruence. Both Mundy and Field count as quantity primitivists because they recognize fundamental mass comparison facts. According to nomic reductionism, on the other hand, there are no fundamental quantity comparisons. Instead, the only physically significant relations among magnitudes concern the roles those magnitudes play in the laws, and in particular facts about the different consequences the magnitudes have for how things move around. According to nomic reductionism we can make sense of the structure of mass solely in terms of the fact that 1000kg mass and 0.01kg mass are associated with different consequences for how things accelerate. For example, objects with a mass of 1,000kg accelerate more slowly in response to a given force, and dispose other things to accelerate faster due to their gravitational attraction. According to nomic reductionism, then, spatiotemporal quantities like acceleration, spatial distance, and temporal duration are special. The world s temporal and spatial arrangement is fundamental, and it gives rise to the structure of physical quantities like mass and charge. I will refer to non-spatiotemporal quantities as physical magnitudes. 4 claims about comparisons among objects, like (1), are prior to claims that attribute intrinsic properties to objects, like the Elephant has a mass of 1000kg. The absolutist must arguably invoke second-order comparative facts like (2), as in Armstrong (1988), Bigelow and Pargetter (1988) and Mundy (1987). So both the absolutist and the comparativist are quantity primitivists since they invoke some kind of fundamental quantity comparisons. 4 Mass was once thought of not as a quantity associated with inertia or gravitation but as measure of how filled in a region of space is. Conceiving of mass in this way would make it a quasi-spatiotemporal quantity. But I take it that the contemporary conception of mass is not like this.

12 7 According to the quantity primitivist, which things are more massive than others is independent of what the laws are like. 5 The nomic reductionist, on the other hand, claims that the elephant has more mass partially in virtue of the fact that it s harder to throw an elephant than an egg. 6 This difference has two important implications for how to think about quantities. First, the quantity primitivist and the nomic reductionist disagree about which features of our numerical representations of quantities are merely conventional and which are physically significant. And the two views have different accounts of the content of the laws. We use real numbers to represent magnitudes of mass. Why is this? According to the quantity primitivist this is because the the mass magnitudes stand in various fundamental ordering and distance relations, and the real numbers also stand in ordering and distance relations, and so we can use this numerical structure to represent the physical structure of the mass magnitudes. Not all features of a numerical scale are physically significant. For example, which mass magnitude gets assigned to the number 1 is purely conventional. But, for example, the ordering in a scale for mass is physically significant because it reflects the ordering relations the mass magnitudes stand in. 7 According to quantity primitivism, quantities like mass are structured independently of the laws. So the primitivist regards laws like f = ma as linking families of properties that 5 By saying that the structure of a quantity is independent of the laws I mean something stronger than modally free, namely that facts about a quantity s structure do not metaphysically depend on, or are metaphysically explained by facts about laws. I take the phrase f obtains in virtue of g to articulate a distinctively metaphysical type of explanation that can also be expressed with a variety of locutions like f because g, f depends on g, for f to obtain just is for g to obtain, or g grounds f. I take the notion to be familiar from various debates in philosophy; for example, Socrates challenge to Euthyphro was to say whether the pious acts are pious in virtue of the love of the gods or vice versa. Similarly, physicalism is the claim that everything obtains in virtue of physical facts. Recently a number of philosophers have explicitly defended the importance of the notion of fundamentality and metaphysical explanation to metaphysics; see, for example, Fine (2001), Schaffer (2009), Rosen (2010), or Sider (2012). But I invite the reader who prefers to understand these debates in terms of supervenience to read my claim as a supervenience thesis as well. 6 One might think that quantity primitivists do not merely deny this in virtue of claim; they endorse an opposing claim, namely that elephants are harder to throw than eggs in virtue of having more mass. But this second in virtue of claim is plausibly a merely physical explanation. The quantity primitivist need not think that facts about throwing elephants are grounded in facts about their masses. Rather, the distinctive claim of quantity primitivism is the denial that mass comparisons are grounded. 7 The details of what it takes for a scale to be faithful depends on the version of quantity primitivism. For instance, on Mundy s (1989) account, a faithful scale assigns real numbers r 1, r 2 and r 3 to mass magnitudes m 1, m 2 and m 3 if and only if (a) r 1+r 2=r 3 if and only if (m 1, m 2, m 3) and (b) r 1 > r 2 if and only if (m 1, m 2). On Field s (1980) account, a faithful scale assigns real numbers r 1, r 2, r 3 and r 4 to objects o 1, o 2, o 3 and o 4 if and only if (a)mass-between(o 1,o 2,o 3) if and only if either r 1 r 2 r 3 or r 1 r 2 r 3 and (b) mass-congruent(o 1,o 2,o 3,o 4) if and only if r 1 r 2 = r 3 r 4.

13 8 have a prior structure. The fact that one mass property m 1 is twice as large as another m 2 is not constituted by or grounded in the fact that anything with m 1 will accelerate half as fast under a given force than something with m 2 is another. Nomic reductionism is the claim that the only physically significant comparisons among physical magnitudes concern the role those magnitudes play in the laws. But there is an apparent problem with this claim, since the laws appear to take quantity structure for granted. f = ma, for example, seems to say that if one mass m 1 is twice as large as another m 2, then a body with m 2 will accelerate twice as fast as one with m 1 under the same force. If the law has this content then it cannot, on pain of circularity, be used to account for what makes m 1 twice as large m 2. In fact this is not a problem for the nomic reductionist, for she claims that the content of the laws is exhausted by specifying what consequences different magnitudes have for how things accelerate. Consider the case of f = ma. Everyone will agree that some features of the statement of this law are purely conventional. For example, the choice to measure mass in kilograms instead of grams does not reflect the fact that there is anything special about objects with a mass of 1 kg. So it is natural to think that stating the law as f = 1000ma with mass measured in grams is physically equivalent to f = ma with mass measured in kilograms. The nomic reductionist claims that the physically significant content of this laws is exhausted by determining which magnitudes of mass and force are associated with which accelerations. Given an appropriate assignment of masses, and forces and accelerations to numbers, this content can be expressed mathematically as f = ma. This encodes the fact that anything with a mass of 3 kg that experiences a resultant force of 6 Newtons accelerates at 2 ms 2. Since the nomic reductionist regards the kilogram scale as a conventional choice, it is helpful to be able to refer to the magnitudes independently of a numerical scale. Let m 1 be the property of having a mass of 3kg and f 1 be the property of experiencing a resultant force of 6 Newtons. Then the statement of f = ma encodes the fact that anything with m 1 and f 1 has an acceleration of 2 ms 2. But this fact can be encoded by different assignments of numbers to mass and force properties if the statement of the law is adjusted, so writing the law as f = ma involves a

14 9 conventional choice on our part. There are other combinations of mathematical scales for mass and mathematical statements of the laws that express this same content. For example, consider the schmilogram scale, the assignment of numbers to mass properties with the feature that if a mass property gets assigned to the number r by the kilogram scale, it gets assigned to 1 r by the schmilogram scale. Pair this with the mathematical statement f = a/m. This combination also entails that anything with m 1 and f 1 has an acceleration of 2 ms 2. 8 So the nomic reductionist will say that f = a/m with mass is measured in schmilograms is equivalent to f = ma with mass measured in kilograms. Nomic reductionism is the claim that there are no fundamental mass comparisons. All God had to do when he created the world was to settle the laws (that is, determine the ties between physical magnitudes and acceleration.) He did not then also have to settle the facts about the structure the mass properties stand in. This contrasts with quantity primitivism, which holds that some mass comparisons are fundamental. To make vivid the difference between quantity primitivism and quantity reductionism, consider a scenario in which (by the quantity primitivist s lights) the mass comparisons are quite different but in which things behave just the way they actually do. Suppose the laws of nature were different so that wherever some quantity actually depends on a thing s mass it instead depends on the inverse of its mass. 9 And further suppose the distribution of masses differs from that in the actual world so that wherever x kg mass is actually instantiated, 1 x kg mass is instantiated instead. In this scenario everything behaves in exactly the same way that it actually does! Everything that actually has a small mass has a large mass in this scenario; but since the laws treat things with large masses the way that our actual laws treat things with small masses, things behave in the same way. This scenario and agrees with the actual world on the trajectory taken by every massive object A mass of 3 kg is equivalent to a mass of 1/3 schmilograms; since in the schmilogram scale a = fm, this means that f = 6.1/3 = 2ms 2. 9 That is, f = a/m instead of f = ma and the gravitational attraction between massive bodies were 1 proportional to m x.m y.d instead of to mx.my. 2 d 2 10 Consider an object o, with mass m 1 kg, d m from the centre of the Earth, which has mass m 2 kg. In w 1 m 2 kg. The force due to gravity on o is actually m 1m 2 1 m 1 o has a mass of kg and the Earth has a mass of 1 the force due to gravity on o in w is 1 1 m 1 = m 1m 2 m2.d 2 d 2 d 2 ;. Assume that the only force on o is that due to

15 10 The quantity primitivist must claim that this is a scenario in which things have different masses than in the actual world. But because everything in scenario has a property that plays the same role in laws as in the actual world, the nomic reductionist claims things have the same mass in the two world. Some might take this to be a count against nomic reductionism, since intuitively the inverse-mass has a different distribution of mass and nomic reductionism fails to vindicate this intuition. I am happy to admit that this is a cost of the view, but one that is outweighed by the arguments in favour of it. Moreover, there is little reason to expect our intuitions about the nature of physical magnitudes to be reliable, and so this is an issue on which our beliefs ought be guided by the arguments rather than by which view is closest to common sense. If nomic reductionism is correct then the laws of nature play a role in accounting for the structure of physical quantities. But it is neutral on which philosophical account of laws of nature is correct. It is therefore compatible with the view that the laws are fundamental sui generis facts, or the view that the laws are explained by the essential dispositions of the fundamental properties, or even, as I argue in section 4, the view that facts about laws are themselves reducible to facts about the distribution of properties over spacetime. I have contrasted two accounts of the structure of physical quantities. Quantity primitivism invokes fundamental quantitative structure to make sense of physical quantities. Nomic reductionism simply locates this structure in the nomic connections among magnitudes. The following section argues that the extra structure of quantity primitivist is explanatorily redundant. Section 3 argues that nomic reductionism is preferable because it affords an explanation for the fact that magnitudes fail to be freely recombinable. 2 The Argument From Redundant Structure This section argues that quantity primitivism is committed to redundant structure. My case against quantity primitivism is analogous to the case against endorsing facts about m 1m 2 gravitation attraction of the Earth. The acceleration of o is actually given by a = f = d 2 m m 1 = m 2. The d 2 acceleration of o in w is given by a = f.m = m 1m 2. 1 d 2 m 1 = m 2, which is precisely what it actually is. d 2

16 11 absolute velocity in the context of Newtonian gravitational mechanics. You are moving at different speeds relative to different things. You are stationary with respect to your armchair, moving at about 66,500 mph around the sun, and at about 515,000 mph around the centre of the Milky Way. But how fast are you really going? Do you also have an absolute velocity in addition to all these relative velocities? The answer depends on how much structure spacetime has. 11 If spacetime has Newtonian structure then, since there is a fact about the spatial distance between any two points, there is a fact about your absolute velocity (just find the distance between the region you are located in now and the region you were located in a moment ago). But if spacetime merely has Galilean structure 12 then there are no facts about the distance between non-simultaneous points, and so there is no such thing as absolute velocity. 13 The consensus among scientists and philosophers of science is that, assuming the laws are those of Newtonian gravitational mechanics, we should think spacetime has only Galilean structure. This conclusion is typically supported by one of two arguments. Both arguments ultimately depend on the observation that facts about absolute velocity are undetectable. As Newton himself was aware, what the laws say about how things in a system interact is completely independent of how fast the system is moving. But this means that even if you have an absolute velocity it s impossible to detect it. What does it take for some quantity q to be detectable? 14 On a natural way of thinking about detectability, there must at least be some measuring procedure for q such that (a) its outputs are reliably correlated with the value of q and (b) its outputs are accessible to us, so that the procedure allows us to form reliable beliefs about the value of q. 15 It s natural to think that if there is such a measurement procedure then the results of a measurement can be recorded with the position of a pointer, or by being written down on a piece of paper, or by being described verbally, or by being displayed on a computer screen. After all, if a 11 As Maudlin (1993) points out the issue of whether there are absolute velicities is independent of whether substantivalism is correct, since the relationist could recognize absolute velocities by, for example, positing distance relations between the temporal parts of material bodies. 12 Sometimes called neo-newtonian spacetime. 13 As Newton s Bucket argument showed, there had better be facts about emphabsolute accelerations. In Galilean spacetime absolute acceleration cannot be defined in terms of absolute velocity. Instead, a basic distinction between accelerating and non-accelerating trajectories is baked in. 14 Thanks to an anonymous referee for urging me to clarify the for of the argument in what follows. 15 This way of thinking about detectability comes from Albert (1996) and Roberts (2008).

17 12 measurement procedure allows me to form reliable beliefs about q then surely I can decide to record my belief by writing it down. Let s say that measurement procedures like this are empirical measurement procedures. 16 That is, if there is an empirical measurement procedure for q, then there must be some condition C (i.e., whatever the set-up conditions for the procedure are) such that if C, for any value of q, x, the laws guarantee that the procedure results in a recording of x only if the value of q is x. 17 But given Newtonian gravitational mechanics it is impossible for there to be an empirical measurement procedure for absolute velocity! Suppose there were such a procedure and that it is carried out by Sally the scientist. Sally writes down the result on a piece of paper: My absolute velocity is 5 mph. Now imagine a world that is just like ours, except that everything is moving 1000 miles an hour faster in a certain direction. The two worlds agree on the relative motions and positions of every object. Therefore Sally writes down My absolute velocity is 5 mph. in this world too. But Sally s absolute velocity is different in the two worlds, and so the measurement procedure must have produced a false result in at least one of them. Therefore the procedure can t have been reliable after all. Thus we have an argument that absolute velocities are undetectable that appeals to the following principle: (P1) A quantity q is empirically detectable in a world w only if there is an empirical measurement procedure for q in w. Since there is no empirical measurement procedure for absolute velocities in NGM, absolute velocities are empirically undetectable. However, this principle is arguably too limited in scope. Consider the hypothesis ( Stationary ) that spacetime has Newtonian structure, and the laws are those of Newtonian gravitational mechanics together with the stipulation that it is a law that the center 16 Even if there is no empirical measurement procedure for some quantity it doesn t quite follow that it is undetectable. Perhaps there could be beings that have the ability to directly sense their absolute velocity, even though they would be in the bizarre position of being unable to communicate their sensations in the form of letters or in spoken conversation or by sign language. (This question is taken up in Roberts (2008).) But I take it to be eminently plausible that we are not like these beings, and so the only quantities detectable to us are those for which there exist empirical measurement procedures. 17 This condition should obviously be relaxed to accommodate statistical errors.

18 13 of mass of the universe is stationary. There is an empirical measurement procedure for absolute velocities given Stationary: to find the absolute velocity of some body, simply find its motion relative to the center of mass of the universe. But there is an important sense in which absolute velocities would still be undetectable given Stationary. For the measurement procedure described above is only a reliable measurement procedure for absolute velocities if the laws are those of Stationary. So our having evidence concerning the absolute velocities of things depends on our having evidence that the laws are those of Stationary. But we don t have any such evidence, since the world according to Stationary is indiscernible from a world in which spacetime only has Galilean structure and the laws are simply those of NGM. 18 The general point is that in order for something to be detectable, not only must there be laws that allow us to construct a certain measuring device, we must also know what the laws are that govern our measuring devices. 19 This suggests that we adopt a more general principle concerning detectability: (P2) If there is a measurement procedure for some quantity q if the laws are L, but not if the laws are L*, and we have no evidence that the laws are L rather than L*, then q is undetectable. This principle correctly predicts that even if Stationary is true, absolute velocities are undetectable. We have looked at two reasons for thinking that absolute velocities are undetectable in NGM. One argument against Newtonian spacetime appeals directly to the fact that positing undetectable structure is a theoretical vice. (D1) Newtonian spacetime requires empirically undetectable structure that is not endorsed by Galilean spacetime. (D2) All else equal, if one theory T 1 posits less undetectable structure than another theory T 2, this a reason to prefer T 1 over T 2. (D3) So, all else equal, we should prefer positing Galilean to Newtonian spacetime. 18 Dasgupta (2013) appeals to this reasoning to argue that absolute mass facts, as opposed to merely mass ratios, are undetectable. 19 Many thanks to an anonymous reviewer for drawing my attention to the importance of this issue.

19 14 Note that this argument is not committed to verificationism; it is compatible with (D2) that we are justified in endorsing theories that posit plenty of undetectable structure. (D2) merely says that when two theories are otherwise equally worthy of belief, we should prefer the one with less undetectable stuff. 20 While I think this argument has some merit, there is a closely related but importantly different argument that appeals instead to redundant structure. One worry with the argument against undetectable structure is that is relies on a false premise, (D2), since there is nothing intrinsically suspect about positing undetectable facts. We should believe the hypothesis that provides the best explanation of our evidence, and this hypothesis may well appeal to undetectable features of the world. Rather, the correct diagnosis for why we shouldn t posit facts about absolute velocities is simply because we should attribute as little structure to the world as we can get away with. And the fact that absolute velocities are empirically undetectable shows that we can get away with attributing less structure to the world than is required by Newtonian spacetime. Absolute velocities aren t needed to make sense of world, and so they are explanatorily redundant. 21 Thus I think that the following argument captures the best case against positing the full structure of Newtonian spacetime in the context of Newtonian gravitational mechanics (NGM): (N1) Galilean spacetime has less structure than Newtonian spacetime. (N2) All else equal, if two theories are both empirically adequate we should prefer the theory that attributes the least structure to the world. (N3) NGM with Newtonian spacetime and NGM with Galilean spacetime are both empirically adequate. (N4) So, all else equal, we should prefer positing Galilean spacetime to Newtonian spacetime. While this argument does not rely on the claim that undetectable structure is itself 20 The case against Newtonian spacetime is framed in these terms by Maudlin (1993) and Dasgupta (2012). 21 The case against Newtonian spacetime is put this way, for example, by Earman (1989), Brading and Castellani (2005), Roberts (2008), North (2009), Baker (2010), and Belot (2011).

20 15 problematic, undetectability considerations play a crucial role in providing a justification for (N3). A world with Galilean spacetime differs from a world with Newtonian spacetime only with respect to features that are undetectable. So if NGM combined with Newtonian spacetime is empirically adequate, so too is NGM combined with Galilean spacetime. If absolute velocities were detectable, then a theory that dispensed with them would be no good. I take something like (N2) to be ubiquitous in both scientific and common sense reasoning, and enshrined in inference to the best explanation. Again, (N2) is not the claim that simpler hypotheses are always better; just that, faced with two hypotheses that are otherwise equally worthy of belief, we should prefer the one that attributes less structure to the world. The claim that Galilean spacetime attributes less structure may be justified in a number of ways. One method is to appeal to a modal test for having more structure: there are a great many distinctions made by Newtonian spacetime that Galilean spacetime ignores, since for every way of arranging things over Galilean spacetime, there are many different arrangements in Newtonian spacetime that agree on the relative motion but not the absolute motions of things. But this modal test is only a rough heuristic. For as Dasgupta (2013) observes, someone might believe that spacetime has Newtonian structure, but also think that the actual world is the only possible world. (Perhaps because he thinks Spinoza was right about modality). In this case there are no more ways of arranging things in Newtonian than Galilean spacetime there is exactly one. But surely this Newtonian s eccentric beliefs about modality does nothing to alter the fact that Newtonian spacetime has more structure than Galilean spacetime. Exactly how to spell out what it takes for one theory to have more structure than another is a vexed question that I won t try to settle here. 22 One thought is that in Galilean spacetime there are no matters of fact about the spatial distance between non-simultaneous points, whereas there are in Newtonian spacetime. Another is that Galilean spacetime has more symmetries than Newtonian spacetime. But I hope it is clear enough that however the notion is spelled out, Newtonian spacetime has more structure than Galilean spacetime. 22 North (2009) contains a detailed discussion of what it means to minimize structure in a physical theory.

21 16 I ve endorsed a particular analysis of why we should reject Newtonian spacetime in the context of Newtonian gravitational mechanics, and I will argue that quantity primitivism should be rejected for similar reasons. But my case against quantity primitivism won t depend on this analysis, for whatever theoretical vice Newtonian spacetime exemplifies it is one that is shared by quantity primitivism. If Newtonian spacetime should be rejected because it contains undetectable structure then, since analogous considerations support the claim that quantity primitivism also contains undetectable structure, quantity primitivism should be rejected on these grounds as well. The quantity primitivist holds that there are primitive facts about which things are more massive than others. I will now argue that facts like these are just like absolute velocities. Worlds that differ only about which things are more massive than others are indiscernible, and so we don t need facts like that to make sense of the world. Quantity primitivism is committed to redundant structure, for the additional fundamental facts it requires perform no explanatory work, and so we should prefer nomic reductionism. This argument mirrors the one given above against Newtonian spacetime. (Q1) Nomic reductionism attributes less structure to the world than quantity primitivism. (Q2) Ceteris paribus, if two theories are both empirically adequate we should prefer the theory that attributes the least structure to the world. (Q3) Nomic reductionism and quantity primitivism are both empirically adequate. (Q4) So, ceteris paribus, we should prefer nomic reductionism to quantity primitivism. This argument is valid, and so it remains only to defend the premises. The premise (Q2) just is the premise (N3) in the argument against Newtonian spacetime, and so I won t say more about it here. As for (Q1), nomic reductionism attributes less structure to the world in the same way that Galilean spacetime posits less structure than Newtonian spacetime. And again, we may appeal to the heuristic that the nomic reductionist ignores distinctions made by the quantity primitivist. But this test suffers from the same limitation as when applied to Newtonian spacetime. A quantity primitivist could deny that the inverse mass world is

22 17 possible, and so deny that that quantity primitivism recognizes any more distinctions than the nomic reductions. (She might think this, for example, because she thought Spinoza was right about modality.) But in this case too, whatever quirky views the quantity primitivist might have about what s possible they surely aren t relevant to the question of how much structure quantity primitivism attributes to the world. Since the quantity primitivist posits extra facts that the nomic reductionist takes to be reducible, I take it to be clear enough that quantity primitivism requires more structure than nomic reductionism. On to (Q3). The case for thinking that nomic reductionism is empirically adequate is also analogous to the case for thinking that Newtonian gravitational mechanics in a Galilean spacetime is empirically adequate. In the case of spacetime, we argued that since NGM with Newtonian spacetime is empirically adequate and NGM with Galilean spacetime agrees in all detectable respects, it too must be empirically adequate. Similarly, I ll argue that since there are no detectable differences between worlds that differ only about which things are more massive than others, nomic reductionism agrees with quantity primitivism in all detectable respects, and so it too is empirically adequate. Why think fundamental quantity comparisons are undetectable? Well, consider whether there is a measurement procedure for mass ordering facts, for example. There must be a procedure that takes two objects, x and y, and results in a recording of x is more massive than y only if x is in fact more massive than y. If the laws are those of Newtonian gravitational mechanics then placing the two objects on a balance and writing down which way the balance tips is a reliable measurement procedure for mass orderings. But now consider the inverse-mass world described in the previous section, in which wherever there is actually something with x kg mass there is something with 1/x kg mass instead; and the laws there are the result of replacing every appearance of m in statements of the actual laws with 1/m. In this world the balance procedure is not a reliable way of measuring mass-orderings. So in order to obtain evidence about mass-orderings we need first to know that the laws are those of the NGM, not those of the inverse-mass world. But we don t have any such evidence! The two sets of laws are equally simple and elegant. And the two worlds agree perfectly on the trajectories things take. But this means that they agree perfectly on where dials in detecting devices point, and on what anyone

23 18 ever wrote down, said or, for that matter, thought. 23 Since we don t have any evidence that the laws are those of NGM and not inverse-ngm, we have no reason to think that balances provide us with evidence about mass orderings. The situation is analogous to the case of Stationary, the hypothesis that spacetime has Newtonian structure but it is a law that the center of mass of the universe is stationary. If Stationary is correct, then there is a measurement procedure for absolute velocities. But absolute velocities are still undetectable, since we could have no evidence that the laws are those of Stationary rather than simply those of NGM. So for every world recognized by the quantity primitivist there is a an empirically equivalent world recognized by the nomic reductionist. Thus quantity primitivism is empirically adequate only if both are. This completes the defense of premise (Q3) in the argument. Even though nomic reductionism attributes less structure to the world than quantity primitivism is still able to account for the data. The extra structure of quantity primitivism is redundant structure. Is it possible the quantity primitivist to respond that there are still primitive mass comparisons that are preserved across the inverse mass world? Perhaps a quantity primitivist could adopt a radical version of comparativism, and claim that the only fundamental mass comparisons are betweenness facts, such as the fact that 2kg mass is between 1kg mass and 3kg mass. Such betweenness facts are preserved under the operating of taking inverses, and so my chosen example does not yet show that this version of quantity primitivist has less structure than nomic reductionism 24 In response, note first just how impoverished this version of quantity primitivism is: it is obviously impossible to capture mass orderings or ratios with just the resources of mass betweenness facts. So this radical comparativist agrees with the nomic reductionist that, for example, writing Newton s second law as f = ma was a conventional choice on our part because it is physically equivalent to f = m/a with mass measured schmilograms. Because this comparativist regards mass betweenness facts as physically real, however, she unlike the nomic reductionist thinks that worlds with a different distribution of 23 Assuming that fixing the configuration of your brain and environment settles the thoughts you have. 24 I thank an anonymous referee for pressing this objection to me.

24 19 mass betweenness facts are genuinely different ways for the world to be. 25 But now the nomic reductionist can argue along precisely the same lines as before that such differences are undetectable, and so we have no need to posit irreducible mass betweenness facts to make sense of the world. 26 This section argued that quantity primitivism attributes unnecessary structure to the world and that we should prefer nomic reductionism instead. But perhaps all else is not equal, and we are justified in positing the extra structure of quantity primitivism because we thereby obtain a theory with greater explanatory power. The next section argues that this is not the case and that nomic reductionism is preferable precisely because of its greater explanation power. 3 The Argument Against Brute Necessary Connections If science is a guide to which properties are fundamental, then the fundamental properties are physical magnitudes, like 1kg mass or 3 Coulombs charge. It is determinate properties like these, rather than determinables like mass or charge, that are fundamental, since settling the distribution of mass and charge underdetermines the distribution of magnitudes of mass and charge. The fact that the fundamental properties are physical magnitudes is puzzling, since physical magnitudes are not freely recombinable: the fact that something has 1kg mass entails that it has no other magnitude of mass. The distribution of one magnitude, say 1kg mass, imposes constraints on allowable distributions of other magnitudes of mass. But we expect the basic building blocks of the world to be, in David Hume s words, entirely loose and separate. 27 Although property incompatibilities do not concern distinct existences since they constrain the properties of a single particular, positing necessary connections 25 For example, consider a world (w cut) in which mass is distributed in the following way: for all objects, o, if o actually has a mass of x kg then (i) if x is greater than 1000 then o has a mass of x kg in w cut; (ii) if x is less than or equal to 500 o has a mass of x kg in w cut; (iii) if x is greater than 500 but less than or equal to 500, o has a mass of x 500 kg in w cut. 26 Could the quantity primitivism admit that there are no mass betweenness facts, but adopt some even sparser conception of which mass comparisons are physically real? It is unclear how she could, for whatever comparisons she regards as real will serve to distinguish worlds the nomic reductionist regards as identical. The nomic reductionist, recall, denies that there are any physically significant mass comparisons, except those that concern how magnitudes are linked with acceleration. 27 David Hume (1975) [1748] p. 61.

25 20 between fundamental properties is a serious theoretical vice. First, necessary connections are evidence of dependence. Entailments between distinct properties for example, between natural and normative properties are usually taken as a strong indication that the properties in question are not both fundamental but instead that one obtains in virtue of the other, or that each obtains in virtue of some further fact. So if a theory posits fundamental properties or relations that stand in necessary connections this is evidence against that theory. Secondly, necessary connections call out for explanation. This is plausibly what drives many to infer from the supervenience of normative on natural properties that both are not fundamental. For if natural and normative properties were both fundamental, it would be mysterious why they were so nicely choreographed. We might imagine that God creates the world, one fundamental property at a time. Once he has settled the distribution of the natural properties, he goes on to specify the distribution of normative properties but does so in precisely such a way that one class of properties supervenes on the other. Why would God s creative powers follow this pattern? Thirdly, in general necessary connections between fundamental properties mean redundancy at the fundamental level. Suppose that property P necessitates property Q, but that P and Q are both fundamental. Now suppose that when God creates the world, He settles all the fundamental facts, one by one. First He decrees that some object o has P, and everything that is required for o to have P appears. Then God decrees that o is also Q and now nothing new happens, because o s having Q was already settled by God s first decree. The fact that o is Q provides no new information about the world since it is entailed by o s having P. The natural conclusion is that Q is not fundamental after all, since this results in a sparser set of fundamental facts that still characterizes reality completely. A preference for sparser accounts of the world militates against theories with redundancy, and therefore against theories according to which there are necessary connections between fundamental facts. Finally, necessary connections between fundamental properties appear to rule out Humean reductionism about laws of nature. Although I take Humean reductionism to be supported by powerful arguments, it is a controversial position, and so this last consideration won t be

26 21 persuasive to everyone. Humean reductionists regard facts about laws and related concepts like dispositions, powers, or causation to reduce ultimately to non-nomic facts. The most promising form of Humean reductionism is the best system analysis defended most notably by David Lewis, according to which the laws are the theorems of the axiomatization of the distribution of properties that achieves an optimal balance of informativeness and simplicity. Does the best system analysis count as a version of Humean reductionism? It does, as long as facts about the distribution of properties are themselves non-nomic. Lewis claimed that the fundamental properties are non-nomic, insofar as they obey a principle of recombination. Lewis offered various formulations of this principle of different strengths. But on the most natural reading, the fundamental properties are freely recombinable as long as the fact that one property is instantiated somewhere has no entailments for where any other property is instantiated. But if the fundamental properties are physical magnitudes as conceived by the quantity primitivist, then this is surely false! Magnitudes in a quantity are incompatible, and so the fact that some object o has 1kg mass does entail something about where other magnitudes of mass are instantiated: that o does not also have 2kg mass. So it would appear that necessary connections between fundamental properties rule out one of the most plausible accounts of laws. 28 I conclude that there are compelling reasons to avoid positing necessary connections between fundamental properties and relations. As I ll argue, however, quantity primitivism, unlike nomic reductionism, requires endorsing problematic brute necessities. Quantity primitivism is the claim that there are fundamental mass comparisons. Recall that a quantity primitivist could hold that either first-order (as in the elephant is more massive than the egg ) or second-order comparisons (as in 2kg mass is greater than 1kg 28 Ned Hall (unpublished manuscript) has recently argued, in effect, that there is nothing problematic about entailments like this. It is part of Humean reductionism about laws that laws reduce to facts that are themselves non-modal. This is supposed to rule out, inter alia, essentialism about physical quantities, the view that properties play their nomic roles essentially. But the Humean cannot say that a property is non-modal as long as it respects a strong form of recombination because physical magnitudes fail such a condition. Hall recommends that the Humean endorse the following version of the claim the laws reduce to non-modal facts: [t]he fundamental ontological structure of the world is given by the distribution of perfectly natural magnitudes in it, where these magnitudes respect an inter-magnitude principle of recombination. All other facts, including facts about the laws, reduce to these facts. This Humean has in some robust sense fewer fundamental modal facts. On the face of it, however, the claim that satisfying this limited recombination principle suffices for being entirely non-modal just looks ad hoc.

27 22 mass ) are fundamental. The first-order quantity primitivist posits fundamental mass comparison relations that hold between objects. But while these relations are supposed to be fundamental they are not freely recombinable. For example, Field (1980) invokes the two relations massbetweenness and mass-congruence. These relations are stipulated to obey certain constraints; for example, mass-betweenness(x,y,z) and mass-between(w,x,z) entail that massbetweenness(w,y,z). Similarly, mass-congruence(x,y,w,z) and mass-congruence(x,y,u,v) entail mass-congruence(u,v,w,z). But this is just to say that these fundamental relations violate our ban on brute necessary connections. The second-order quantity primitivist posits fundamental mass comparisons among properties. For instance, Mundy (1979) posits two second-order relations, and, where intuitively (p 1, p 2 ) means p 1 is less than or equal to p 2, and (p 1, p 2, p 3 ) means that p 3 is the sum of p 1 and p 2. The second order quantity primitivist must also recognize brute necessary connections. The following question arises for the second-order quantity primitivist: is it essential to a magnitude of mass that it stand in the mass comparisons it actually does? Take two mass magnitudes, m 1 and m 2, and suppose that (m 1, m 2 ). Does it follow that in every world in which they are instantiated, (m 1, m 2 )? Suppose that it does, so that our quantity primitivist has an essentialist second-order account. Then her theory entails that m 1 and m 2 are necessarily incompatible. But as I argued, since m 1 and m 2 are fundamental properties we expect them to be recombinable, and so this is a reason to reject essentialist second-order quantity primitivism. But adopting a non-essentialist version of second-order quantity primitivism instead is little help since it requires brute necessary connections of a different form. Suppose that second-order mass comparisons are not essential to the magnitude they relate. If so then the second-order quantity primitivist is free to regard first-order properties as being freely recombinable after all. Suppose that the predicate has 1kg mass refers to the property m 1 and has 2kg mass refers to the property m 2. Since the fundamental properties are freely recombinable, there is a world something has both m 1 and m 2. But this does not mean that the quantity primitivist must claim that it is possible for an object to have both 1kg

28 23 mass and 2kg mass! That is, this quantity primitivist can distinguish the following claims: (3) Nothing can have both 1kg mass and 2kg mass. (4) Nothing can have both m 1 and m 2. Our primitivist denies (4). But she can still endorse (3), on the grounds that has 1kg mass and has 2kg mass do not rigidly designate m 1 and m 2. The primitivist could instead offer a semantics for these predicates so that they only refer to a pair of properties if they stand in the appropriate mass comparisons. The quantity primitivist would then have an explanation of (3) on the basis that it is analytically true, and yet deny (4), and so avoid positing necessary connections among fundamental properties. But while some brute necessary connections are avoided using this strategy, others are not. For suppose the primitivist regiments the structure of mass properties in terms of Mundy s relations and. While the non-essentialist allows that m 1 and m 2 are recombinable because they are only contingently related by, the primitivist must still endorse the following brute necessities: (5) necessarily, for any properties p 1, p 2, p 3 if (p 1, p 2 ) and (p 2, p 3 ) then (p 1, p 3 ). (6) necessarily, for any properties p 1, p 2 if (p 1, p 2 ) then nothing has both p 1 and p 2. So even if the non-essentialist quantity primitivist can explain magnitude incompatibilities, she still posits necessary constraints associated with second-order mass comparisons like (5) and (6). 29 So all versions of quantity primitivism involve positing necessary constraints that govern the fundamental properties or relations. This is a reason to avoid quantity primitivism if we can. 29 Could the quantity primitivist claim that it is merely a contingent fact that behaves like an ordering relation, and so deny (5) and (6)? This would make it mysterious what the content of the quantity primitivist s view is; it amounts to the claim that there are some second-order relations that happen to behave in a certain way in the actual world. But it is unclear what these relations are and what they have to do with the structure of mass: merely labeling a relation or calling it less-than-or-equal-to on its own does not explanatory work. Perhaps a version of quantity primitivism could be developed that escapes these problems, but the ball is certain in the quantity primitivist s court.

29 24 The nomic reductionist does not require positing any special necessary connections. For one, the nomic reductionist denies that there are any fundamental mass comparisons and so avoids the necessary connections associated with them like (5) and (6). The nomic reductionist also has a natural way to account for magnitude incompatibilities. The nomic reductionist holds that the only physically significant quantity comparisons concern which properties are associated with which acclerations. Again, the nomic reductionist could develop an essentialist or a non-essentialist version of her view. The non-essentialist nomic reductionist can, like the non-essentialist quantity primitivist, hold that the fundamental properties are freely recombinable because they play their role in the laws contingently. On this view, it is a contingent fact that the properties m 1 and m 2 play the role of 1kg mass and 2kg mass in the laws, and so it is possible that something instantiate both m 1 and m 2, as long as they play different roles in the laws. And again, the nomic reductionist can offer a semantics for has 1kg mass so that it is analytic that nothing has both 1kg mass and 2kg mass. According to the essentialist nomic reductionist, what it is to be the 1kg mass property is to play a certain role in the laws; that is, to have certain consequences for how things move around. This means that in order for a thing to instantiate multiple magnitudes from the same quantity it would have to be disposed to follow incompatible trajectories. It is part of playing the 1000 kg mass role that if one object has 1000 kg mass, then a second massive object separated from it by 1 mm and under no other influences accelerates at 66.7 mm s 2. And it is part of playing the 2000 kg mass role that if one object has 2000 kg mass, then a second massive object separated from it by 1 mm and under no other influences will accelerate at twice that rate, 133 mm s 2. So if an object has both 1000kg mass and 2000kg mass, then nearby massive objects under no other influences accelerate at 66.7 mm s 2, and they accelerate at 133 mm s 2. Since this is impossible, we have an explanation for why nothing can have more than one magnitude of mass. Of course, it might seem that all that has been achieved is that the bulge has been moved in the spatiotemporal carpet, for surely it is just as mysterious that nothing can have two different magnitudes of acceleration as it is that nothing can have two magnitudes of mass!

30 25 The nomic reductionist has two responses to this worry. The nomic reductionist could accept that in order to explain magnitude incompatibilities she must simply take for granted that nothing can have more than one acceleration. Even if this were so, the nomic reductionist would have the advantage over the quantity primitivist of having reduced the number of unexplained necessary connections: instead of being burdened with physical magnitude incompatibilities as well as spatiotemporal incompatibilities, she need appeal only to the former. And of course, this response leaves open the possibility that spatiotemporal incompatibilities may themselves be explained. But the nomic reductionist needn t be this concessive. Even without assuming that nothing can accelerate at multiple rates, there are independent reasons to think that there could not be laws that require things have more than one rate of acceleration. First, laws of temporal evolution describe how the states evolve over time as a function of how objects and properties are distributed. If things have multiple accelerations then as a result they must end up in multiple locations. But the causal influences on some object depend on its location. For example, if two objects are both 1m and 2m apart, do they experience the forces they would experience if they were 1m apart or the force they would experience if 2m apart? If both, then the object that already has two locations must accelerate in two different ways, and so it must have even more locations in the future; and every object that interacts with the multiply located object must itself have multiple accelerations and therefore multiple locations. So it is somewhat implausible that there could be laws like this. Secondly, even if laws with multiple outputs were coherent, if laws of temporal evolution are to be fully general they must generate a unique output. For example, consider the law f = ma. On a natural way of regimenting this law, it says that for any object, the unique resultant force on it is equal in magnitude to the product of its unique mass and its unique acceleration. Suppose the law were weaker and simply required that for any object, it has a mass m i, a resultant force f i and an acceleration a i such that f i =m i a i. This law leaves it open that there are other triples of mass, force and acceleration that do not satisfy f = ma. So if laws are to be fully general they must give unique outcomes for how states evolve over time.

31 26 If this is right then laws of temporal evolution arguably do not merely require that states evolve in a certain way; they also entail that any other way the states evolve violates the laws. Therefore it is a consequence of the laws that if one object has 1000kg mass then any massive object at a distance of 1 mm has exactly one acceleration, namely, 66.7 mm s 2. The same goes for 2000kg mass. And it is a straightforward logical impossibility for the unique acceleration of some object o to be 66.7 mm s 2 and for the unique acceleration of o also to be 133 mm s 2. But there is nothing mysterious about necessary connections that are logical connections, such as the fact that it is impossible for something to exist and not exist at the same time, or that it is impossible to be both square and not square at the same time. The nomic reductionist, therefore, can explain magnitude incompatibilities without having to invoke brute necessary connections. This a compelling reason to prefer nomic reductionism. 4 Objections Anticipated In this section I will respond to what I take to be the most pressing objections to nomic reductionism. The first objection is that nomic reductionism is not compatible with Humean reductionism about laws of nature. According to Humeanism reductionism, facts about laws are grounded in the global distribution of properties. The most promising story about how facts about laws are so grounded is the best system account associated with John Stuart Mill and Frank Ramsey and developed by David Lewis, according to which the laws are the axioms of the systematization of the facts that achieves the best possible balance of informativeness and simplicity. 30 Humean reductionism is motivated by a desire to reduce all nomic facts to non-nomic facts, like facts about how properties are distributed over spacetime. For this reason Humeans typically claim that the fundamental properties are themselves freely recombinable, in the sense that the fact that one property is instantiated at one location places no constraints on where any other property is instantiated. This provides a clear 30 The loci classici are Mill (1973), Ramsey (1970), and Lewis (1973).

32 27 sense in which the facts about the distribution of properties are non-modal. Now, discussion about the merits of Humean reductionism has largely ignored the fact that the fundamental properties are clearly not freely recombinable in this sense, for they are physical magnitudes, and physical magnitudes in the same family are incompatible! So not only is nomic reductionism compatible with Humean reductionism, it provides the best way for the Humean reductionism to uphold the free recombination claim, thereby ensuring that the reduction is to wholly non-nomic facts. The Humean nomic reductionist takes the distribution of determinate properties over spacetime as fundamental, but does not recognize any fundamental facts about how these determinate properties are structured, or even about which properties are members of the same family. According to nomic reductionism comparisons among properties including facts about which properties are members of the same family are grounded in the role of those magnitudes in the sparse dynamical laws. And the nomic reductionist is free to claim that what makes the sparse dynamical laws laws is simply that they are part of the systematization that achieves a best balance of informativeness and simplicity. The sparse dynamical laws contain less structure than the richer laws recognized by the quantity primitivist. But the Humean mosaic recognized by the nomic reductionist has less structure too, since she does not regard any facts about comparisons among properties as fundamental like the quantity primitivist does. So I see no reason to suspect that the best system analysis should be adequate only on the assumption that quantity primitivism is correct. 31 The second objection concerns the explanation of magnitude incompatibilities. I have so far ignored what might seem to be the most natural account of quantities, which holds that quantities are functions from objects to numbers. On this view, if I have a mass of 75kg, this is because the mass-in-kilograms function maps me to the number 75. Call this the Pythagorean view. The Pythagorean view captures the structure of quantities in a simple and direct way. And since part of what it is to be a function that it has a unique output, 31 This position resembles in some respects the view proposed by Ned Hall (unpublished ms.), according to which the Humean mosaic only includes facts about the trajectories things take. Facts about physical magnitudes are not fundamental. Instead, they are made true by the fact that the best system of these trajectories attributes physical magnitudes to material objects together with laws about how things move around given their properties. The proposal in the text differs in that the Humean mosaic, the facts being summarized, also includes which properties things have. But it is similar in emphnot including facts about the structure those properties stand in, such as facts about which things are more massive than others.

33 28 it would seem that there is no mystery as to why nothing can have two distinct masses at the same time. For all its appeal, the Pythagorean view faces formidable problems and should be rejected. The Pythagorean view either requires implausibly privileging a choice of scale or is committed to massive redundancy in the fundamental facts. Take the fact that I have a mass of 75kg. According to the Pythagorean view this is so in virtue of the fact that the mass-in-kilograms function maps me to the number 75. But the fact that we chose to measure masses with the kilogram scale, as opposed to the grams scale or the solar masses scale was an arbitrary decision on our part. I have a mass of 75,000 grams, so the mass-in-grams function maps me to the number 75,000. Are both mass functions fundamental, or is one privileged? It is hugely implausible that we have just happened to hit upon the one that is fundamental, since there is nothing physically special about objects that have 1kg mass. But if instead each mass function that corresponds to a choice of scale is fundamental, then the view requires a vast proliferation of fundamental facts. On the amended version of the view it is a fundamental fact that the mass-in-kilograms function maps me to one number, and it is a fundamental fact the mass-in-grams function maps me to another number, and so on. Another family of problems with the Pythagorean View is that it is not an intrinsic feature of an object that a certain function maps it to one number rather than another. If the fundamental properties are quantities, and quantities are just functions from objects numbers, we are left with a picture of the world on which nothing has any interesting intrinsic nature. But surely the world is a certain way intrinsically. Moreover, it is one of our most fundamental convictions about properties and explanation that things behave the way they do at least partially because of the way they are intrinsically. Balls roll because they re spherical, for example. But if the fundamental facts about the nature of things only concern functions from objects to numbers, then we lack this form of explanation for how things behave. I conclude that the Pythagorean view is not an attractive way to explain magnitude incompatibilities.

34 29 5 Conclusion I conclude that we should adopt nomic reductionism over quantity primitivism. We should prefer a picture of the world that is not unduly mysterious nor unnecessarily complex; quantity primitivism fails on both counts. The arguments presented in this paper generalize to other disputes in metaphysics. Most obviously, many of the same considerations given here arise for spacetime as well, and I believe this means we should conclude that facts about the spatial or temporal separation of two points is reducible to facts about potential causal interactions between those points. Another promising application for the argument against redundant structure is in the debate over quidditism, the issue of whether the identity of a property is fixed by its nomic role, or haecceitism, the thesis that an object s identity is not fixed by its qualitative features. It has been a mistake to see the principal obstacle to quidditism or haecceitism as revolving around the problem of whether we can come to know quiddistic or haecceitistic facts; a more compelling objection is that these views are committed to explanatorily redundant structure.

35 30 3. The Causal Theory of Spacetime. We naturally think that the way things are arranged in space and time is a fundamental feature of the world. This paper argues that this is a mistake, and instead defends the causal theory of spacetime, the view that facts about the spatial and temporal distance between material bodies reduce to facts about how they interact. This is a radical claim, for the vast majority of philosophers at implicitly accept spacetime primitivism, the claim that the spatial and temporal arrangement of the world is irreducible whether this arrangement is to be understood in terms of relations between material objects or in terms of the structure of substantival spacetime. The causal theorist, on the other hand, holds that the spatiotemporal arrangement of the world reduces to facts about lawful dependence. On this view, all God had to do when he created the world was determine how the inhabitants of the world interact and the spatiotemporal arrangement of the world emerged from this basis. I present three arguments in favor of the causal theory. First, if the spatiotemporal arrangement of the world were independent of its causal structure then it would be empirically inaccessible. I ll argue that this means that we don t need irreducible facts about space and time to make sense of the world they are explanatorily redundant. Second, if spacetime primitivism is correct then we must give up a plausible minimality constraint on the fundamental primitive spacetime facts are metaphysically redundant. Third, adopting the causal theory of spacetime allows us to explain why spatial and temporal relations fail to be freely recombinable. Spacetime primitivists must instead posit unexplained necessary connections between the basic spatiotemporal relation. As I ll explain this is a form of axiomatic redundancy. Here s the plan for the paper. Section 1 develops and clarifies my thesis. Section 2

36 31 presents the argument from explanatory redundancy, section 3 presents the argument from metaphysical redundancy, and section 4 presents the argument from axiomatic redundancy. Finally, sections 5 and 6 describe how the causal theory may be combined with both non- Humean accounts of laws of nature and Humean reductionism. 1 Causal Structure How strongly things interact depends on how far apart they are. An erupting volcano in the Galapagos Islands might hurt some turtles there but will probably do little harm to turtles in Japan. Given spacetime primitivism this is something that merely happens to be true given the form the laws take. But according to the causal theory, facts about causal influence are constitutive of distance in space and time; it is partly in virtue of the fact that the Japanese turtles would be less harmed by the volcano that they are farther away. For the spacetime primitivist it is one thing to say that two things are far apart, and quite another to say how they causally interact. It is therefore coherent to suppose, if spacetime primitivism is correct, that the facts about how far apart things are might vary independently of the facts about causal influence. Suppose that in the actual world Billy smashes a window by throwing a brick at it: Actual World: Billy throwing the brick caused the window to smash. If spacetime primitivism is correct then we can keep this causal fact fixed while varying the spatiotemporal arrangement of the world. For example, consider a world in which everything is half its actual size.

37 32 Halved World (w half ): w half is just like the actual world except that every material object is half as large as it actually is, but the laws are also scaled down so that things interact just as they actually do. 1 A smaller Billy throws a smaller brick and smashes a smaller window. The ratios among distances are unchanged in w half. Although Billy is half his actual size, his size relative to the window is unchanged. The spacetime primitivist could hold that only distance ratios are physically real, and conclude that since w half agrees on all the distance ratios it is merely a rediscription of the actual world. 2 But if facts about distance are independent of facts about causal interaction then we can consider worlds in which things causally interact just as they actually do but in which distance ratios are not preserved. Of course, since these worlds have different spatiotemporal arrangements there is a clear sense in which things don t causally interact the way they actually do. But I take it that there is an equally intuitive sense in which the following worlds describe worlds with the same causal relations. Shrinking World (w shink ): In w shrink Billy is half his actual size at t 1. But the next second, at t 2, he is half as large as he was at t 1. The next second his size has halved again. Everything is constantly shrinking in size, but the strength with which things interact weakens to compensate perfectly. 1 That is, replace every occurrence of d, x, v, and a in the laws by d/2, x/2, v/2 and a/2. 2 Shamik Dasgupta (2013) argues that since halving every distance would not make any detectable difference to the world, facts about absolute distances are empirically redundant and this is a reason only to regard facts about distance ratios as physically real.

38 33 Cut and Paste World (w paste ): In w paste Billy is not located in front of the window. However, the region he occupies, r 2, stands in all the relations of causal dependence that r 1, the region that is actually occupied by Billy, stands in. So when Billy throws the brick, it emerges from r 1 as if Billy were located there. 3 It s easy to multiply examples. w grow is just like the actual world except that everything doubles in size every second but the strength with which things interact weakens to compensate perfectly. The shards of glass produced are much larger than in actuality, although they are no more dangerous. w faster is just like the actual world except that the time between events decreases, so that everything happens faster and faster. Billy is able to escape the crime scene much sooner, although he s no more likely to evade capture. And so on. These scenarios differ radically about the spatial and temporal arrangement of the world. But it is clear that they have something in common; in a quite intuitive sense, things causally interact just as they actually do. And they agree not only about actual causation but also about causal dependence. For example, in each world if Billy were to throw a feather at the window the window would fail to break, and if he were to throw a grenade it would smash the window more thoroughly. I ll say that these worlds have the same causal structure. The causal theory of spacetime is the claim that the spatiotemporal arrangement of the world reduces to its causal structure. As I ll make clear, I understand this an explanatory claim: the causal theorist claims that spacetime structure obtains in virtue of causal structure. 4 For example, it is partly because 3 David Albert (1996) considers a similar scenario to argue that the geometrical appearances are accounted for by the dynamical laws. 4 The claim that spacetime is not fundamental (or emergent ) comes up in discussions of two ares of physics. David Albert (1996) defends an interpretation of Bohmian mechanics on which the fundamental space is an extremely high-dimensional configuration space. Similarly, the idea that four-dimensional spacetime is not fundamental is a feature of some versions of string theory. But both of these claims involve taking some kind of spatial or geometrical structure as fundamental, and so I will treat these claims as versions of spacetime primitivism.

39 34 a nuclear explosion in the vicinity of my coffee mug would cause me to die, while a nuclear explosion on the moon would cause me no harm, that I am closer to my coffee mug than to the moon. Like many philosophers, however, I do not take causation to be fundamental, so I will characterize causal structure in terms of the laws of nature. If the causal theory of spacetime is correct then these scenarios do not, after all, correspond to different possibilities. Is this a count in its favor, because intuitively these scenarios involve making distinctions without a difference? Or does this count against the causal theory because the causal theory fails to recognize intuitively distinct possibilities? I m not sure; I find my intuitions to pull in both directions. So one response is to say with David Armstrong, spoils to the victor! and conclude that we should let our intuitions be guided by theory, not vice versa. But I am happy to grant that there is some weak intuitive pressure against the causal theory. Intuitions about the nature of fundamental reality must sometimes be revised in the face of countervailing evidence, as with the appearance that the sun revolves around the Earth, or that some events are objectively simultaneous. Similarly I will argue that the evidence from intuitions against the causal theory is outweighed by the arguments in favor of the view. The issue at stake between the causal theory of spacetime and spacetime primitivism is independent of the dispute between substantivalism and relationism about spacetime. Substantivalism is the claim that spacetime regions or points do not depend for their existence on material objects. 5 According to relationism, spatiotemporal relations only hold between material objects and claims about spacetime itself are derivative relative to claims about spatiotemporal relations between material objects. 6 This is a dispute about which kinds of entities instantiate fundamental spatiotemporal properties and relations: material objects or spacetime points and regions? The question at issue in this paper cross-cuts the substantivalism-relationalism debate since it concerns the spatiotemporal relations themselves, not their relata. If substantivalism is correct, we may ask whether it is a fundamental 5 I don t think the dependence at issue is merely modal dependence, since in principle a substantivalist could deny that facts about spacetime are modally independent of material bodies. Rather, substantivalism properly construed is the claim that spacetime points and regions do not exist in virtue of material objects and their properties. See Dasgupta (2011). 6 I ll frame the discussion in terms of spatial and temporal relations, but I intend this to be neutral about whether nominalism or realism about properties and relations is correct.

40 35 fact that two points are one meter apart or whether this obtains in virtue of other facts. And if relationism is correct, we may ask whether it is a fundamental fact that two material objects are one meter apart, or whether this obtains in virtue of other facts. I will assume substantivalism for the sake of presentation but the extension to relationism should be straightforward. The causal theory of spacetime is the claim that spacetime structure reduces to causal structure. In what sense do the scenarios above describe worlds with the same causal structure? In each world, Billy s throwing a brick causes a window to break. But we cannot explain what the worlds have in common by saying that a duplicate of Billy causes a duplicate of the window to break, since duplication is standardly defined so that two objects are duplicates only if their parts stand in the same spatial relations. 7 Let us use a slightly different notion instead. Say that two objects are purely qualitative duplicates if and only if their parts have the same perfectly natural properties. 8 Objects can be purely qualitative duplicates even though they are not duplicates; one window may be a purely qualitative duplicate of another even though it s twice as large. I ll say that purely qualitative duplicates have the same purely qualitative profile. We can extend this notion to spacetime regions: the purely qualitative profile of a spacetime region r is given by saying which perfectly natural properties are instantiated at each subregion of r. 9 In the actual world, Billy s throwing the brick caused the window to break. In w half smaller Billy caused a smaller window to break. Billy and smaller Billy are purely qualitative 7 For example, see Lewis (1986) p. 60. Lewis requires that if objects are duplicates then their parts must stand in the same perfectly natural relations. I should note that I am happy to take mereological relations among regions as primitive. 8 That is, there is a one-one mapping between their parts such that every part of one object is mapped to a part of the other object with the same perfectly natural properties. (Note that everything is a part of itself). The restriction to perfectly natural properties is required so that objects can be purely qualitative duplicates even though they fail to share properties like being exactly ten feet from President Obama. 9 Three remarks. First, while I am assuming substantivalism I shall try to remain neutral about supersubstantivalism, the thesis that material objects are identical to spacetime regions. So I will remain neutral on whether the perfectly natural properties are instantiated merely at spacetime regions (by being instantiated by a material object located there) or by the regions themselves. I assume that spacetime is continuous. Second, spacetime is not made up of enduring points, but instantaneous points-at-a-time, events. Spacetime regions perdure: they exist at multiple times by having parts at those times. So the notion of location appealed to in a claim like the ball started at l, bounced off the wall and returned to l encodes information about the relation between two regions that make up l: the spacetime region initially occupied by the ball, l i, and the region occupied later l f. Third, if spacetime is discrete then some worlds that differ by a scale factor will not count as having the same causal structure. For example, if spacetime is a lattice made of points one unit apart, then objects can differ in size by being different finite numbers of units across. Therefore no discrete spacetime has the same causal structure as any continuous spacetime.

41 36 duplicates, and so too are the broken window and the smaller broken window in w half. So the regions occupied by Billy in the two worlds (r Billy and r SmallBilly ) are purely qualitative duplicates, as are the regions occupied by the broken windows (r W indow and r SmallW indow ). But these two pairs of regions have something else in common: they stand in the same relations of causal dependence. For example, if r Billy contained an exploding bomb, this would cause r W indow to contain a more thoroughly smashed window. And the same is true of r SmallBilly and r SmallW indow. I ll say these regions are causally similar. We can characterize causal similarity in terms of the what laws say about how the regions interact. I write L(p) to express the fact that it is a consequence of the laws that p. Suppose that the laws entail that if there is an explosion of a certain magnitude sufficiently close to a glass window then that window breaks. In particular suppose that if the laws entail that r Billy contains an exploding bomb then r W indow contains a thoroughly smashed window. Then we can write this as L(BOMB(r Billy ) SMASH(r W indow )). Since we are interested in causally similar regions in worlds with different spatiotemporal arrangements we will focus on lawful conditionals that do not presuppose facts about how things are arranged in space and time. We can do this by selecting just those conditionals that describe the interaction between regions in non-spatiotemporal terms. For example, if p B and p S are the purely qualitative profiles that the regions r Billy and r W indow would have if they contained an exploding bomb and a smashed window, then L(p B (r Billy ) p S (r W indow )) characterizes the dependence between the regions without presupposing spacetime structure. I will say that these lawful conditionals that ignore spacetime structure express direct dependence relations among regions. 10 Causal similarity can now be defined in terms of relations of direct dependence. Two regions are causally similar when they have the same forward-looking causal profile and the same backward-looking causal profile. Two regions have the same forward-looking causal profile (Causal Profile F ) if and 10 It is worth noting that this lets us distinguish two different senses in which a window can be said to be broken. In one sense (the spatial sense) a window is broken if and only if its proper parts are no longer in spatial contact. But in the other sense (the causal sense) a window is broken if and only if its parts no longer compose a causally cohesive object, so that, e.g. pushing one part causes the whole window to move. These senses can come apart imagine a version of the cut and paste world, but swapping out a section of an unbroken window instead. In this new cut and paste world the window is spatially but not causally broken.

42 37 only if what goes on in the regions has the same affects on other regions: Causal Profile F : two regions, r 1 in w 1 and r 2 in w 2, have the same forwardlooking causal profiles if and only if for any purely qualitative profiles p 1 and p 2, if there is a region r 3 such that L(p 1 (r 1 ) p 2 (r 3 )) then there is a region r 4 such that L(p 1 (r 2 ) p 2 (r 4 )). 11 Similarly, two regions have the same backward-looking causal profile (Causal Profile B ) if and only if they are affected in the same way by what is going on in other regions: Causal Profile B : two regions, r 1 in w 1 and r 2 in w 2, have the same backwardlooking causal profile if and only if for any purely qualitative profiles p 1 and p 2, if and only if there is a region r 3 such that L(p 1 (r 3 ) p 2 (r 1 )) then there is a region r 4 such that L(p 1 (r 4 ) p 2 (r 2 )). If two regions are both causally similar and purely qualitative duplicates, I will say that they are causal duplicates. Now we can characterize causal structure in the following way: Causal Structure: two worlds w 1,w 2 have the same causal structure if and only if there is a one-one mapping between regions in w 1 and regions in w 2 that maps every region to a causal duplicate region. This captures what is shared between the worlds we looked at previously. In each world, we can identify regions that are causal duplicates of actual regions. For example, the region that actually contains Billy is a causal duplicate of the smaller region in w half that contains the smaller Billy. 11 To incorporate chancy causation we need to require that if r 1 having p 1 lawfully entails that the chance that r 3 has p 2 is c, then r 2 having p 1 lawfully entails that the chance that r 4 has p 2 is c.

43 38 Causal duplicate regions in w actual and w half. We are now in a position to state the causal theory of spacetime: The Causal Theory: facts about the world s spatiotemporal structure obtain in virtue of its causal structure. Since the causal structure of the world is determined by the laws of nature, there is an obvious problem with this claim. The laws as we know them make claims about spatiotemporal structure: they say, for example, how things accelerate under given forces, and how gravitational attraction varies as a function of the distance between two massive bodies. So we might worry that since the laws themselves appeal to spatiotemporal notions, we cannot use the laws to analyze spacetime itself. Precisely what fundamental reality is like according to the causal theory and therefore how the causal theorist responds to this problem depends on which account of the laws of nature is correct. The most promising version of reductionism about laws is the best system analysis defended notably by David Lewis. 12 On this view, a statement counts as a law when it encodes a lot of information in a particularly efficient way. Compare all the ways of summarizing the contingent facts about the world. Some are very informative, but extremely complicated: we could just list every property instantiated by each spacetime point. Others are extremely simple, but not very interesting: we could simply state the number of material particles that exist. According to the best system analysis, the laws of nature are those statements that together achieve the best balance of informativeness and simplicity. 12 See Lewis (1983). Lewis was building on the regularity accounts of John Stuart Mill and Frank Ramsey.